Question

⦁ Use a paragraph, flow chart, or two-column proof to prove that "ZX" is the perpendicular bisector of side "WY".

Answer 1

Answer with Step-by-step explanation:

We are given that  two triangles ZXW and ZXY.

$\angle WZX\cong YZX$

$ZW\cong ZY$

We have to prove that Zx is perpendicular bisector of WY.

By two column proof

Proof:

Statement :$\angle WZX\cong \angle YZX, ZW\cong ZY$

Reason: Given

Statement:$ZX\cong ZX$

Reason: Reflexive property of congruence

Statement:$\triangle WZX\cong \triangle YZX$

Reason:SAS postulate

Statement:$WX\cong XY, \angle ZXW\cong \angle ZXY$

Reason: CPCT

$\angle ZXW+\angle ZXY=180^{\circ}$

Reason: Linear pair

Statement:$\angle ZXW+\angle ZXW=180$

Reason:Substitution property

Statement:$\angle ZXW=180$

Reason: addition property

Statement:$\angle ZXW=\frac{180}{2}=90^{\circ}$

Reason: Division property of equality

Statement:$\angle ZXW=\angle ZXY=90^{\circ}$

Reason: Substitution  property

When  a line segment is a perpendicular bisector then it divides the opposite side into two equal parts and it is perpendicular to opposite side.

Hence, ZX is a perpendicular bisector of WY

Answer 2

Explanation:

The given congruences are corresponding sides and angles of ΔWZX and ΔYZX. Segment ZX is congruent to itself, so ΔWZX ≅ ΔYZX by SAS.

Then ∠WXZ ≅ ∠YXZ and WX ≅ YX by CPCTC. The latter means X is a midpoint and that ZX is a bisector of WY. The former means the angles of the linear pair ∠WXZ and ∠YXZ must both be 90°, as they are both congruent and supplementary.

Therefore, ZX is perpendicular to WY and also its bisector.